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${:[3x^(2)+6x-24=0],[(3x^(2)+6x)/(x)=24]:} {:[3(x-2)(x+4)],[3x-6+3x+12]:}$

$\begin{array}{l}3 x^{2}+6 x-24=0 \\ \frac{3 x^{2}+6 x}{x}=24\end{array}$ $\newline$ $\begin{array}{l}3(x-2)(x+4) \\ 3 x-6+3 x+12\end{array}$

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Set-builder notation

Full solution

Q.

\begin{array}{l}3 x^{2}+6 x-24=0 \\ \frac{3 x^{2}+6 x}{x}=24\end{array}

\newline

\begin{array}{l}3(x-2)(x+4) \\ 3 x-6+3 x+12\end{array}

Factorize first equation: First, let's look at the first equation $3x^2 + 6x - 24 = 0$ . We can try to factor this.
Find factors: To factor, we look for two numbers that multiply to $(3 \times -24) = -72$ and add to $6$ .
Write factored equation: The numbers that work are $12$ and $-6$ because $12 \times -6 = -72$ and $12 + (-6) = 6$ .
Set factors equal: Now we can write the equation as $3(x - 2)(x + 4) = 0$ .
Solve for x: Setting each factor equal to zero gives us the solutions $x - 2 = 0$ and $x + 4 = 0$ .
Simplify second equation: Solving these, we get $x = 2$ and $x = -4$ .
Divide by $x$ : Now let's look at the second equation $(3x^2 + 6x) / x = 24$ . We can simplify this by dividing each term by $x$ .
Subtract $6$ : This gives us $3x + 6 = 24$ .
Divide by $3$ : Subtracting $6$ from both sides gives us $3x = 18$ .
Find $x$ : Dividing both sides by $3$ gives us $x = 6$ .

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